On the elastic stability of peridynamic material models in two-dimensional deformations

Peridynamics has emerged as a powerful nonlocal framework for modeling material behavior, particularly in the context of damage evolution and fracture mechanics. Building upon Silling’s energy minimization criterion for assessing the stability of peridynamic correspondence materials, this work extends Hill’s stability criterion to the nonlocal framework, and establishes a generalized energy-based stability criterion for peridynamics. Based on this criterion, we propose a novel and concise method for verifying the material stability of linearized peridynamic models even without requiring material isotropy. For isotropic peridynamic materials, we conduct a rigorous examination of the linearized displacement field under finite deformation. From this analysis, we derive fundamental conditions for linear stability and prove several key theorems revealing : (1) the fundamental role of Poisson’s ratio in determining stability; (2) that linear stability is independent of the influence function; and (3) that linear stability depends exclusively on the singular values of the deformation gradient. We demonstrate that the proposed stability criterion is fully characterized by the positive definiteness of a specific tangent modulus tensor, which enables stability analysis via its eigenvalues. By applying the Sylvester criterion, we precisely delineate the stability region in deformation gradient parameter space and systematically investigate its parametric dependence on Poisson’s ratio. Our theoretical framework reveals a fundamental dichotomy: materials with low Poisson’s ratios are more prone to instability under shear deformations, whereas those with high Poisson’s ratios are more susceptible to instability under volumetric compression. These theoretical predictions are systematically validated through computational experiments, demonstrating strong agreement between analytical results and numerical simulations. This work not only deepens the fundamental understanding of stability in peridynamic material models but also delineates the applicability limits of peridynamics under finite deformation, offering valuable insights for the development of robust constitutive models in future peridynamic research.